>>16652363
>>16652372
If you want a system with a least two different numbers such that at least one of them is non-zero, you can't allow division by zero in the usual ring inverse of multiplication sense.
>why can't, for example, [math]\frac{x}{0}=\pm{\infty}[/math]?
Because, even in nonstandard analysis, where there are non-zero infinitesimals and infinitely large numbers, [math]\left|\frac{1}{\textrm{Large}}\right|[/math] still isn't [math]0[/math] but a strictly greater than [math]0[/math] infinitesimal and [math]\left|\frac{1}{\textrm{Infinitesimal}}\right|[/math] still isn't [math]+{\infty}[/math] but an infinitely large number strictly less than [math]+{\infty}[/math]. That is, if you divide by non-zero infinitesimals the answer is infinitely large, but never the largest because there is no smallest non-zero infinitesimal. Allowing division by zero, the answer you get isn't the largest infinitely large number because no longer there are infinitesimals or infinities or most of the classical numbers anymore: to remain consistent, the answer is zero and every number in the system is zero.

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why is dividing by 0 not allowed but dividing by 2 allowed? why is the number 0 so special?